Question

3. Continuous dynamical systems - Dimension 2 (a) Suppose the ODE system describes a continuous dynamical system in two dimensions (here f: R2 + R and g: R² R are two functions with smooth partial derivatives). Draw the corresponding vector field in the case that f(x,y) = x2 - y2 8(x, y) = x+y+1

Answer 1

fage f giver dynamical system of non linear system (von diverse differential equations) dr -y? f(x,y) dy = x+y+1 - 91*.v) - 0 - © to draw the vector fields of the above system. Step : fixed points f(x, J) = 0 g(x, y) =0 à - y² = 0 = x=+y at J+ 1 = 0 = y + y + 1=00 24 + 1 =0 - -> y=-=- = y = -1 , - 2) is the only critial point.

Step ④: Linearization of the system near the count point. Jl") were u-x-(-2) - o-y- (-2) Jacobian matsins of f(y), yw.yo whene s at o critical printa y 10 - - 24 7 I ff If a Jord y , I dg dg Ә - 1 (-,-2) 1) f-1 17 - or Linearize system : Stef & - ( JIC) Eigen values of to J-d I/=0 where B 11--[57]) I I loo] =) - =0 1 1-d! (1-d) (1-d) - 120 - #d-1-o =) = d? 220 = -)

: Sino both & of the Cref - behavion systems Since the eigen values die se, da = -52 are positive and negative red members. hence the system is unstable about the critical point (1, 2). Such a critical point is called saddle point. Step 6: Eigen vectors: when dia 52 then eigen vector when dears then eigen vector (-1 + 2 v = V2 = NY So the general solution of the system Jgt -1t 12 ce (1 ) cze -5-) + Goat / call Cuct) Jt ( a K-1 + 5a) e et + Ca(-12-1) e 1(*) Godt +₂ě Jot

exef l vector kaze fields graphs the about (-1,2) field vectors but direction to draw the bat (-1,0) en 1915) * This in this region, vector fields are moving from (-2/2 ) take the point (2.0) the ( !] ( 4 ) = 1 - 2 that is vector fields are morning from (110) towards (2 Now take (0, 1) then 1 1 1 1 1 2 ] = { 1 ] Pie Lo, by then 1 :1-1 ) vector fields graphi: JUT